Yohji Akama, Bobo Hua, Yanhui Su, Haohang Zhang
On one hand, we study the class of graphs on surfaces, satisfying tessellation properties, with positive Forman curvature on each edge. Via medial graphs, we provide a new proof for the finiteness of the class, and give a complete classification. On the other hand, we classify the class of graphs on surfaces with positive corner curvature.
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We provide an overview of Forman-Ricci curvature and persistent homology, and how their combination can be applied to the study of networks. We discuss how the usually employed augmented Forman-Ricci curvature formula, only valid for quasiconvex augmented networks, can be extended to the non-quasiconvex case. We apply three versions of quasiconvex Forman-Ricci curvature (plain, triangle-augmented, and pentagon-augmented) to build time filtrations on non-quasiconvex networks, both model and real-world. Our results suggest that triangle-augmented curvature should be used until the non-quasiconve…
Vertex evaluation of multiplex graphs using Forman Curvature
本文提出利用Forman曲率对多乘图进行顶点评估，通过实验验证其在识别网络中心顶点和分类网络结构方面的有效性。
Forman--Ricci Curvature on Contact-Sequence Temporal Networks via Spatiotemporal Prism Complexes
本文提出一种几何框架，通过棱柱构造将接触序列时间网络提升为真正的单纯复形，并开发了Forman-Ricci曲率，对比了两种变体，证明其在均匀权重下一致，并通过数值实验验证了理论预测。
We introduce Forman-Ricci curvature and its corresponding flow as characteristics for complex networks attempting to extend the common approach of node-based network analysis by edge-based characteristics. Following a theoretical introduction and mathematical motivation, we apply the proposed network-analytic methods to static and dynamic complex networks and compare the results with established node-based characteristics. Our work suggests a number of applications for data mining, including denoising and clustering of experimental data, as well as extrapolation of network evolution.
Augmentations of Forman's Ricci Curvature and their Applications in Community Detection
本文研究了Forman Ricci曲率的增强（AFRC），从理论和实验上证明其能提供网络结构洞察以用于社区检测，且计算成本低于基于ORC的方法，新算法性能与ORC方法相当。
Network analysis using Forman curvature and Shapley values on hypergraphs
本文提出一种结合Forman Ricci曲率与Shapley值的定量度量——组合评估，通过证明其性质并在具体图上与传统中心性比较，验证了该方法的实用性。
Markov and lattice bases for Forman-Ricci curvature of graphs
本研究发展图的Forman-Ricci曲率相关马尔可夫基的代数组合理论，证明不可或缺马尔可夫移动度数随图最大度数二次增长，用三度马尔可夫移动构造格基并结合强化学习寻找特定图的马尔可夫移动。
Edgewise Envelopes Between Balanced Forman and Ollivier-Ricci Curvature
本文提出一种方法，通过构建懒惰运输包络并增强Jost和Liu界，建立参数化于2跳局部图组合学的确定性界，将边状评估复杂度从最优运输线性规划降至O(最大顶点度数^1.5)，并验证了这些界在随机图和实网络中的有效性。
Efficient set-theoretic algorithms for computing high-order Forman-Ricci curvature on abstract simplicial complexes
针对高阶 Forman-Ricci 曲率计算成本过高的问题，本文提出集合论表述以揭示瓶颈并加速单纯复形上的计算，同时给出伪代码、FastForman 实现和基准比较。
Discrete Curvatures and Convex Polytopes
研究了凸多胞体骨架上的Forman--Ricci曲率和有效电阻曲率，推导了平均边曲率恒等式，证明了不同维度下正曲率多胞体的存在性与有限性，揭示了正曲率结构的罕见性与约束性。
Efficient Decomposition of Forman-Ricci Curvature on Vietoris-Rips Complexes and Data Applications
该研究针对Vietoris-Rips复形，提出支持局部计算的Forman-Ricci曲率分解及高效局部计算算法，为其几何计算开辟新途径，凸显统计模式支撑几何的重要性。
The combinatorial Ricci curvature of Forman, which is defined at the edges of a CW complex, and which makes use of only the face relations of the cells in the complex, does not satisfy an analog of the Gauss-Bonnet Theorem, and does not behave analogously to smooth surfaces with respect to negative curvature. We extend this curvature to vertices and faces in such a way that the problems with combinatorial Ricci curvature are mostly resolved. The discussion is stated in terms of ranked posets.
This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably, we find a relation to a number of well-established discrete curvatures (Ollivier, Forman, combinatorial curvature) and show evidence for convergence to continuous curvature in the case of Euclidean random graphs. Being both efficient to calculate and highly amenable to theoretical analysis, these resistance curvatures have the potential to shed new light on …
Exploring the space of graphs with fixed discrete curvatures
本文研究具有指定离散边曲率的图构造问题，提出进化算法采样近似曲率的图，并利用 Markov 基理论给出 Forman-Ricci 曲率下精确重建的有限重连移动集合。
Traditionally, network analysis is based on local properties of vertices, like their degree or clustering, and their statistical behavior across the network in question. This paper develops an approach which is different in two respects. We investigate edge-based properties, and we define global characteristics of networks directly. The latter will provide our affirmative answer to the question raised in the title. More concretely, we start with Forman's notion of the Ricci curvature of a graph, or more generally, a polyhedral complex. This will allow us to pass from a graph as representing a …
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